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Full text of "Scientific Papers - Vi"

152      THE  CORRECTION TO  THE  LENGTH  OF TERMINATED  RODS,  ETC.
as the expression for the capacity which is to be made a minimum. paring  the terms in Hf, we see that the two last, corresponding -negative side, vanish in comparison with the other in virtue of the denominator Iog2^a.    Hence approximately
rr_
and (37) becomes
4.
I
2 log b/a    fog2 b/a ~ »J? (kb]    log2 bfa ^868 J? (k,b)......v
when made a minimum by variation of Hl. Thus the effect of the corr depending on the introduction of Hl is simply to wipe out the initial of the series which represents the first approximation to the correction.
After this it may be expected that the remaining terms of th< approximation to the correction will also disappear. On examinatio: conjecture will be found to be verified. Under each value of k in (16 that part of B is important for which h has the particular value wh nearly equal to k. Thus each new H annuls the. corresponding mem' the series in (39), so that the continuation of the process leaves us wr first term of (39) isolated. The inference is that the correction t capacity vanishes in comparison with 6 -T- log2 b/a, or that 81 vanishes in parison with b ~ log b/a. It would seem that 8£ is of the order b -r- lo but it would not be easy to find the numerical coefficient by the p: method.
In any case the correction 81 to the length of the rod vanishes i electrostatical problem when the radius 'of the rod is diminished w: limit—a conclusion which I extend to the vibrational problem specif the earlier portion of this paper.                      lii ----  /(< ---   7    Tr~77 7 \  i         7        .